Shortages of highly-skilled workers in the Science, Technology, Engineering and Maths (STEM) sector have been widely reported in the US and Western Europe for almost a century. Our research examined the best available UK data on the occupational destinations and career trajectories of both graduates and non-graduates. It found no evidence of a shortage of STEM graduates, nor any of the common indicators associated with labour shortages. Only a minority of STEM graduates ever work in the STEM sector, perhaps suggesting that either the work is unattractive or employers only recruit from certain groups. If STEM employers are concerned about recruiting highly-skilled workers, they may need to rethink what they offer to graduate recruits in terms of 'quality of work', and to also reconsider the extent to which higher education can be expected provide ‘ready-made’ employees that suit their specific needs.